A customer FlexClient page that is accessing the server through Citrix will open the login page and accepts the certificate to proceed to the password page but when the password is entered the page returns back to the Username login page. When I open Flex Client on the server everything opens fine.

See if your customer gets a prompt like the one in this stackexchange question below.  If they do, they will want to clear their browser cookies for the site and when prompted to select a cert hit “Cancel” instead.  

https://superuser.com/questions/1688418/what-is-the-criteria-on-which-chrome-shows-available-certificates-for-client-aut

The flex client web application asks for client certificates to authenticate for “Identified Flex clients”.  When the browser gets asked for a client certificate and finds one that pop-up appears, but if one is chosen and Identified Flex client is not configured the authentication fails in a similar way you described.